Analysis and Probability Seminar

The seminar is joint for the division of Analysis and Probability and its main themes are Mathematical Physics, Probability Theory and Harmonic and Functional Analysis. The seminar is encouraged to be aimed at a broader audience, but it may be of a more specialised nature when indicated in the last line of the abstract. The talks in the seminar are usually 60 minutes, but can also be in the format 2x45 minutes when indicated in the abstract.

At the moment, the seminar takes place via Zoom. See schedule below for links to Zoom meetings.

Should you have any questions or suggestions, feel free to email one of the organizers Martin Hallnäs (Mathematical Physics), Erik Broman (Probability Theory) or Magnus Goffeng (Harmonic and Functional Analysis).

2020-05-19 at 10:00
Jules Lamers (University of Melbourne)

Spin-Ruijsenaars, q-deformed Haldane--Shastry and Macdonald polynomials
The q-deformed Haldane-Shastry model is a partially isotropic (XXZ-like) long-range spin chain that enjoys quantum-affine (really: quantum-loop) symmetries at finite size. I will start by introducing the model and its special properties. Next I will explain how its hierarchy of commuting Hamiltonians are obtained by 'freezing' the spin-version of Ruijsenaars--Macdonald operators. I will also present an explicit expression for the model's exact eigenvectors, which are of ('pseudo' or 'l-') highest weight in the sense that, in terms of the operators from the monodromy matrix, they are eigenvectors of A and D and annihilated by C. Each has a simple component featuring the 'symmetric square' of the q-Vandermonde times a Macdonald polynomial -- or more precisely its quantum spherical zonal special case. If time permits I will outline how we obtain these eigenvectors using the affine Hecke algebra.

This is based on joint work with Vincent Pasquier and Didina Serban (IPhT CEA/Saclay), arXiv:2004.13210 and ongoing.

Participate in the seminar via Zoom:

Previous seminars

2020-01-28 (N.B. MVL14)
Peter Sjögren (Chalmers/GU)

Sharp estimates for the heat kernel in compact rank-one symmetric spaces
We prove sharp two-sided estimates for the heat kernel in any compact Riemannian symmetric space of rank one. As examples of such spaces, we will consider the real and the complex hyperbolic spaces. When the arguments of this heat kernel consists of two almost antipodal points, the kernel has a special behaviour. This phenomenon was found earlier by the authors in the case of the Euclidean sphere. We give similar estimates for heat kernels in the unit ball and for those associated with classical Jacobi expansions. The Jacobi case is the key to the proofs of the results.

This is joint work with A. Nowak and T.Z. Szarek, Wroclaw.

No knowledge of symmetric spaces is required to follow the talk, but it is good to know roughly what a Riemannian manifold is.

On solitons and Calogero-Moser-Sutherland systems
I present a novel soliton equation which is related to a particular integrable system of Calogero-Moser-Sutherland (CMS) type.

My aim is to provided background and motivation to our results for mathematicians with interest in integrable systems and mathematical physics (starting with Russell’s first observation of solitary waves in 1834).

Based on: (in collaboration with Bjorn Berntson and Jonatan Lenells at the Mathematics Department KTH).

2020-02-19 (Wednesday, 10.00, MVL15)
Tuomas Hytönen (University of Helsinki)

Commutators and Jacobians
Let T be a singular integral operator and let b be a function that we identify with a pointwise multiplication operator. We discuss the problem of L^p-to-L^q boundedness of the commutator [b,T] = bT-Tb and its interesting connection -- in the less studied situation when p > q -- to a certain nonlinear PDE, the prescribed Jacobian problem.

Some familiarity with the following notions will be helpful: L^p and first order Sobolev spaces; Hölder/Lipschitz-continuous functions; functions of bounded mean oscillation (BMO) and its duality with the real-variable Hardy space; Calderon-Zygmund singular integral operators (or at least the Hilbert transform).

2020-02-25 (13.15-1500)
Christian Voigt (University of Glasgow)

Graphs, quantum graphs, and their associated C*-algebras
Quantum graphs are certain noncommutative objects which generalize (finite) classical graphs. They have featured recently in connection with the graph isomorphism game in quantum information. In this talk I’ll review some of this background and then present a definition of “quantum graph C*-algebras”, in analogy with (and generalizing) the well-known C*-algebras associated with (finite) directed graphs. Joint work with M. Brannan, K. Eifler and M. Weber.

The first part of the talk requires no prerequisites and the second part is aimed at people with with background in functional analysis/operator algebras

Axel Flinth (Chalmers/GU)

Sparse (and Beyond Sparse) Recovery
Consider a system of linear equations with more variables than equations. As any undergrad students can tell you, there is no hope to solve this system in a unique fashion. Compressed sensing challenges this paradigm. By only assuming that the solution v0 is sparse, it can be found already when the number of equations is basically proportional to the number of non-zero elements. In this talk, we will give a brief overview of the cornerstones of the theory of compressed sensing. If time allows, we will get into a specialized sparsity model where one can prove that recovery is possible for a larger class of systems.

The talk is aimed at a general math audience. In particular, no prerequisite knowledge about compressed sensing is required.

2020-03-09 (Monday 1315, MVL15)
Jani Virtanen (University of Reading)

Entanglement entropy in quantum spin chain models
Abstract: I discuss entanglement entropy of bipartite systems using the von Neumann entropy as measure of entanglement. Some of the most widely studied systems include one-dimensional quantum critical systems, such as quantum spin chains, which in their simplest setting consist of N spins. Of particular interest is the XX spin chain model with zero magnetic field and the study of the von Neumann entropy of the subsystem $P$ of spins on lattice sites $\{1,2,\dots,m\}\cup\{2m+1,2m+2,\dots, 3m\}$, which can be analysed using certain integral representations. For a single block subsystem, the integral representation involves Toeplitz determinants and the entropy can be calculated using the Fisher-Hartwig asymptotic expansion of these determinants. In this talk, we consider a subsystem that consists of two blocks of spins separated by one spin and compute the mutual information between the two intervals using the Riemann-Hilbert approach. Joint work with György Gehér (Reading University) and Alexander Its (Indiana University-Purdue University Indianapolis). The talk is aimed at a general mathematical audience (familiar with basic complex analysis).

Lashi Bandara (University of Potsdam)

Boundary value problems for general first-order elliptic differential operators
The index theorem for compact manifolds with boundary, established by Atiyah-Patodi-Singer in 1975, is considered one of the most significant mathematical achievements of the 20th century. An important and curious fact is that local boundary conditions are topologically obstructed for index formulae and non-local boundary conditions lie at the heart of this theorem. Consequently, this has inspired the study of boundary value problems for first-order elliptic differential operators by many different schools, with a class of induced boundary operators taking centre stage in establishing non-local boundary conditions. 

The work of Bär and Ballmann from 2012 is a modern and comprehensive framework that is useful to study elliptic boundary value problems for first-order elliptic operators on manifolds with compact and smooth boundary. As in the work of Atiyah-Patodi-Singer, a fundamental assumption in Bär-Ballmann is that the induced operator on the boundary can be chosen self-adjoint. All Dirac-type operators, which in particular includes the Hodge-Dirac operator as well as the Atiyah-Singer Dirac operator, are captured via this framework.

In contrast to the APS index theorem, which is essentially restricted to Dirac-type operators, the earlier index theorem of Atiyah-Singer from 1968 on closed manifolds is valid for general first-order elliptic differential operators. There are important operators from both geometry and physics which are more general than those captured by the state-of-the-art for BVPs and index theory. A quintessential example is the Rarita-Schwinger operator on 3/2-spinors, which arises in physics for the study of the so-called delta baryons. A fundamental and seemingly fatal obstacle to study BVPs for such operators is that the induced operator on the boundary may no longer be chosen self-adjoint, even if the operator on the interior is symmetric. 

In recent work with Bär, we extend the Bär-Ballmann framework to consider general first-order elliptic differential operators by dispensing with the self-adjointness requirement for induced boundary operators. Modulo a zeroth order additive term, we show every induced boundary operator is a bi-sectorial operator via the ellipticity of the interior operator. An essential tool at this level of generality is the bounded holomorphic functional calculus, coupled with pseudo-differential operator theory, semi-group theory as well as methods connected to the resolution of the Kato square root problem. This perspective also paves way for studying non-compact boundary, Lipschitz boundary, as well as boundary value problems in the L^p setting.

Published: Sun 10 May 2020.